Our free Quadratic Equation Solver instantly solves any quadratic equation of the form:
ax² + bx + c = 0
Enter the values of a, b, c, and the calculator will:
Show the discriminant (Δ)
Find the roots/solutions (real or complex)
Identify the nature of the roots (distinct, repeated, or complex)
Display the quadratic formula used step by step
Quadratic equations are essential in math, physics, engineering, finance, and computer science. Solving them manually can be time-consuming — especially when working with decimals or complex roots.
With this quadratic calculator you can:
✅ Solve equations step-by-step
✅ Handle both real and complex roots
✅ See the discriminant (Δ) explained
✅ Verify your homework answers instantly
✅ Practice algebra without mistakes
The quadratic formula is:
x = [-b ± √(b² – 4ac)] / 2a
If b² – 4ac > 0 → two real and distinct roots
If b² – 4ac = 0 → one real repeated root
If b² – 4ac < 0 → two complex roots (with “i”)
The term inside the square root (b² – 4ac) is called the discriminant (Δ). It determines the nature of the solutions.
Example 1: x² – 5x + 6 = 0
Δ = 25 – 24 = 1 → Two real roots: x = 2, x = 3
Example 2: 2x² + 4x + 2 = 0
Δ = 16 – 16 = 0 → One real repeated root: x = -1
Example 3: x² + x + 1 = 0
Δ = 1 – 4 = -3 → Complex roots: x = -0.5 ± 0.866i
Q: What is a quadratic equation?
A quadratic equation is a second-degree polynomial in the form ax² + bx + c = 0, where a ≠ 0.
Q: Can the calculator solve complex roots?
Yes ✅ — if the discriminant is negative, it shows solutions in the form p ± qi.
Q: Does this tool show steps?
Yes — it displays the quadratic formula used to reach the solutions.
Q: Can I use this for factoring?
Yes — when the roots are integers, you can rewrite the quadratic in factorised form (x – r1)(x – r2).
Q: What if a = 0?
Then the equation becomes linear (bx + c = 0), and the calculator will not treat it as quadratic.